262 ON THE REFLEXION AND REFRACTION OF LIGHT. 



In like manner, if in the second medium we make 



^ ' (15), 



' dy dx j 

 we get to determine <f> t and ^r y the equations 



(16), 



and as we suppose the constants A and B the same for both 

 media, we have 



7 = . 

 % 9, 



For the complete determination of the motion in question, it 

 will be necessary to satisfy all the conditions due to the surface 

 of junction of the two media. But, since w = and w t = 0, also, 

 since u, v, u t , v t are independent of z, the equations (5) and (6) 

 become 



A + - 



dy) dy 



du . dv __ du / dv, 



dy dx dy dx J 



c/ iJ 



provided x 0. But since x = in the last equations, we may 

 differentiate them with regard to any of the independent varia- 

 bles except x, and thus the two latter, in consequence of the two 

 former, will become 



du _ du j dv dv t 

 dx dx 9 dx dx' 



Substituting now for u, v, &c., their values (13) and (15), in 

 <t> and -Jr, the four resulting conditions relative to the surface of 

 junction of the two media may be written, 



